Stochastic optimal control problems with delays in the state and in the control via viscosity solutions and an economical application

Filippo de Feo

Published 2023-08-28Version 1

In this manuscript we consider optimal control problems of deterministic and stochastic differential equations with delays in the state and in the control. First we prove an equivalent Markovian reformulation on Hilbert spaces of the state equation. Then, using the dynamic programming approach for infinite-dimensional systems, we prove that the value function is the unique viscosity solution of the infinite-dimensional Hamilton Jacobi Bellman equation. Finally we apply this result to a stochastic optimal advertising problem with delays in the state and in the control.